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Theorem cleljust 2114
Description: When the class variables in definition df-clel 2890 are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 2106 with the class variables in wcel 2105. (Contributed by NM, 28-Jan-2004.) Revised to use equsexvw 2002 in order to remove dependencies on ax-10 2136, ax-12 2167, ax-13 2381. Note that there is no disjoint variable condition on 𝑥, 𝑦, that is, on the variables of the left-hand side, as should be the case for definitions. (Revised by BJ, 29-Dec-2020.)
Assertion
Ref Expression
cleljust (𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem cleljust
StepHypRef Expression
1 elequ1 2112 . . 3 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
21equsexvw 2002 . 2 (∃𝑧(𝑧 = 𝑥𝑧𝑦) ↔ 𝑥𝑦)
32bicomi 225 1 (𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wex 1771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1772
This theorem is referenced by:  dfclel  2891
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